GCD in $\mathbb{Z}$ vs $\mathbb{R}[X]$

Common structure. Both $\mathbb{Z}$ and $\mathbb{R}[X]$ are Euclidean domains: for $a,b$ with $b\neq0$, there exist $q,r$ with $a=bq+r$ and $\nu(r)<\nu(b)$ for some size function $\nu$ (absolute value in $\mathbb{Z}$, degree in $\mathbb{R}[X]$).

Bézout. The Euclidean algorithm produces $u,v$ with $au+bv=d=\gcd(a,b)$ in both rings.

Differences. In $\mathbb{Z}$, $\gcd$ is positive by convention; in $\mathbb{R}[X]$, $\gcd$ is monic by convention. Units differ: $\{\pm1\}$ vs $\mathbb{R}^*$.

Conclusion. The Euclidean algorithm and Bézout's identity hold in any Euclidean domain, of which $\mathbb{Z}$ and $\mathbb{R}[X]$ are examples.